1,001 Calculus Practice Problems

Part I

The Questions


Chapter 15

Improper Integrals and More Approximating Techniques

The problems in this chapter involve improper integrals and two techniques to approximate definite integrals: Simpson's rule and the trapezoid rule. Improper integrals are definite integrals with limits thrown in, so those problems require you to make use of many different calculus techniques; they can be quite challenging. The last few problems of the chapter involve using Simpson's rule and the trapezoid rule to approximate definite integrals. When you know the formulas for these approximating techniques, the problems are more of an arithmetic chore than anything else.

The Problems You'll Work On

This chapter involves the following tasks:

·        Solving improper integrals using definite integrals and limits

·        Using comparison to show whether an improper integral converges or diverges

·        Approximating definite integrals using Simpson's rule and the trapezoid rule

What to Watch Out For

Here are a few pointers to help you finish the problems in this chapter:

·        Improper integrals involve it all: limits, l'Hôpital's rule, and any of the integration techniques.

·        The formulas for Simpson's rule and the trapezoid rule are similar, so don't mix them up!

·        If you're careful with the arithmetic on Simpson's rule and the trapezoid rule, you should be in good shape.

Convergent and Divergent Improper Integrals

964–987 Determine whether the integral is convergent or divergent. If the integral is convergent, give the value.

964. 9781118496718-eq15001.eps

965. 9781118496718-eq15002.eps

966. 9781118496718-eq15003.eps

967. 9781118496718-eq15004.eps

968. 9781118496718-eq15005.eps

969. 9781118496718-eq15006.eps

970. 9781118496718-eq15007.eps

971. 9781118496718-eq15008.eps

972. 9781118496718-eq15009.eps

973. 9781118496718-eq15010.eps

974. 9781118496718-eq15011.eps

975. 9781118496718-eq15012.eps

976. 9781118496718-eq15013.eps

977. 9781118496718-eq15014.eps

978. 9781118496718-eq15015.eps

979. 9781118496718-eq15016.eps

980. 9781118496718-eq15017.eps

981. 9781118496718-eq15018.eps

982. 9781118496718-eq15019.eps

983. 9781118496718-eq15020.eps

984. 9781118496718-eq15021.eps

985. 9781118496718-eq15022.eps

986. 9781118496718-eq15023.eps

987. 9781118496718-eq15024.eps

The Comparison Test for Integrals

988–993 Determine whether the improper integral converges or diverges using the comparison theorem for integrals.

988. 9781118496718-eq15025.eps

989. 9781118496718-eq15026.eps

990. 9781118496718-eq15027.eps

991. 9781118496718-eq15028.eps

992. 9781118496718-eq15029.eps

993. 9781118496718-eq15030.eps

The Trapezoid Rule

994–997 Use the trapezoid rule with the specified value of n to approximate the integral. Round to the nearest thousandth.

994. 9781118496718-eq15031.eps with n = 6

995. 9781118496718-eq15032.eps with n = 4

996. 9781118496718-eq15033.eps with n = 8

997. 9781118496718-eq15034.eps with n = 4

Simpson's Rule

998–1,001 Use Simpson's rule with the specified value of n to approximate the integral. Round to the nearest thousandth.

998. 9781118496718-eq15035.eps with n = 6

999. 9781118496718-eq15036.eps with n = 4

1000. 9781118496718-eq15037.eps with n = 8

1001. 9781118496718-eq15038.eps with n = 4